## Birthday Math

Today was Katie’s birthday so we had a birthday-related topic.  Also, to make it feel more special, we had the lesson in the kitchen instead of the usual room.  We agreed that we would do a bit of math and then have cake and tea.

The topic was cutting cakes.  I printed out pieces of paper with some circles drawn on them to do lots of experiments.  The first question was “how many cuts do we need to make to cut the cake into 2 pieces?”  This was very easy; the girls immediately came up with the answer, and each drew a line through one of their circles.  Naturally, we then went on to three pieces.  It was agreed upon that two cuts were necessary and these were the pictures that they drew:

Next, I asked whether one could make two cuts and divide the cake into some number of pieces different from 3.  This again didn’t pose any difficulties, and the girls independently drew the correct configurations for dividing the cake into 4 pieces using two cuts.

The interesting part of the lesson came when we started considering making three cuts.  I thought that the configurations for 4 and 6 pieces would be easy and then I would have to help them with 5 and 7.  However, to my surprise, Katie’s first attempt separated the cake into 7 pieces!  I don’t think that it was actually deliberate, and partially came about out of carelessness.  She was having some trouble counting the parts so I told her to label them with numbers.

Here are  Varya’s first two divisions with three cuts:

Katie also came up with the more traditional way of getting 6:

So at this point we had ways of dividing into 4, 6, and 7 parts (some in multiple ways!).  I asked them whether there was some other number that we could get.  The girls made a few more attempts, but kept getting numbers that we already had.  They also would occasionally draw more than 3 lines, at which point I’d have to point out to them that this was not allowed.  At some point Varya was very close to getting 5, so I helped her a little bit to produce

The girls weren’t nearly as excited by this development as I was, but I was still very happy with the outcome (or rather the process) of our mini-lesson.  And now it was time to actually eat the real cake rather than cut up abstract ones!

As a challenge to the reader, how can one cut a cake into 8 pieces with 3 cuts?